Abstract
Abstract This paper considers the Cauchy problem of solutions for a class of sixth order 1-D nonlinear wave equations at high initial energy level. By introducing a new stable set we derive the result that certain solutions with arbitrarily positive initial energy exist globally.
Highlights
In this paper, we consider the Cauchy problem for the following -D nonlinear wave equation of sixth order: utt – auxx + uxxxx + uxxxxtt = fx, (x, t) ∈ R × (, ∞), ( . )u(x, ) = u (x), ut(x, ) = u (x), x ∈ R, where f (u) = b|u|p, b > and p > are constants, u (x) and u (x) are given initial data, and a > is a given constant satisfying certain conditions to be specified later.When Rosenau [ ] was concerned with the problem of how to describe the dynamics of a dense lattice, he discovered Equation ( . ) by a continuum method
In this paper we intend to extend the existence of global solutions in [ ] with arbitrarily positive initial energy
By using the potential well method [ – ] and introducing a new stable set we show that if the initial data satisfy some conditions, the corresponding local weak solution with arbitrarily positive initial energy exists globally
Summary
1 Introduction In this paper, we consider the Cauchy problem for the following -D nonlinear wave equation of sixth order: utt – auxx + uxxxx + uxxxxtt = f (ux)x, (x, t) ∈ R × ( , ∞), The authors in [ ] first considered the Cauchy problem for Equation ). By the contraction mapping principle, they proved the existence and the uniqueness of the local solution for the Cauchy problem of Equation
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